Commonality in Liquidity in Pure Order-Driven Markets

Transcription

1 Commonality in Liquidity in Pure Order-Driven Markets Wolfgang Bauer First draft: March 31st, 2004 This draft: June 1st, 2004 Abstract This paper extends previous research on commonality in liquidity to pure limit order markets. Using data over three months on the order book of 19 stocks traded at the Swiss Stock Exchange (SWX), we perform a principals components analysis and find evidence of the existence of three to four common factors. The fraction of the variation in liquidity explained by these common factors is higher than what has been found by other studies for quote driven markets, such as Chordia et al. (2000a) or Hasbrouck and Seppi (2001). This fraction of variation of liquidity explained by common factors varies throughout the trading day. It is found that individual stocks liquidity is sensitive to market liquidity and market volatility, as measured by realized volatility. We wish to thank the National Centre of Competence in Research Financial Valuation and Risk Management (NCCR FINRISK) for financial support; NCCR FINRISK is a research programme by the Swiss National Science Foundation. We gratefully acknowledge helpful comments from Rajna Gibson. 1

2 1 Introduction Recent research in finance has focused on the question whether there exist common factors that jointly affect liquidity in different assets. The empirical evidence on the existence of such systematic factors is ambiguous. While Chordia et al. (2000a), Huberman and Halka (2001) and Amihud (2002) find supportive evidence, Hasbrouck and Seppi (2001) find that the common factors in liquidity are relatively small. This paper has two main goals: Firstly, we use data that give more precise information on liquidity on the market to investigate the issue of commonality. Instead of the quote driven markets which have been studied before, we use data on a limit order market. Most of previous empirical work on commonality in liquidity has so far focused on markets in which market makers guarantee a minimal liquidity, in addition to the liquidity provided by limit order books. Therefore, the proxies for liquidity used in these studies are mostly related to the best quoted prices and quantities and actual trades. Recently, data on fully automated markets has become available on which liquidity is solely provided by participants placing limit orders. Research on the existence of commonality in order driven markets is interesting for the following reasons: On one hand, data on the limit order book provide much finer information on the liquidity offered on the market. In addition to the best prices and quantities offered and the actual trades, the traders intention to trade can be observed. In view of the ambiguous evidence on the existence of commonality, the goal of this paper is to investigate whether the additional information provided by the order book yields support for the existence of common factors. And on the other hand, since the sources of commonality of liquidity are not clear, it is far from obvious that the (in addition mixed) results on commonality can be extended from quote driven to limit order markets. Yet, this is an important issue given the large number of stock exchanges that are organized as pure limit order markets and recent considerations by the NYSE to reduce the importance of market makers in favor of an electronic limit order system. In addition, this is the first study on commonality in liquidity using data on the Swiss stock exchange (SWX). Based on the supportive evidence that we find for the existence of common factors, our second goal is to identify these common factors. We therefore try to find some financial variables that affect liquidity of all individual stocks. Previous academic research on commonality in liquidity has been based on partial information on the supply of liquidity: For quote driven markets, only the quoted bid and ask prices and quantities could be used (Chordia et al., 2000a, Hasbrouck and Seppi, 2001, Huberman and Halka, 2001). Hansch (2003) had only access to the best fifteen orders and, in addition, the market he studies 2

3 is not a pure order driven market on its own, but functions in addition to the largest quote driven market of the world, the NYSE. We have access to the complete history of all limit and market orders submitted as well as trades carried out on stocks over a three months period at the Swiss Stock Exchange (SWX) which is a purely order driven market (Ranaldo (2000, 2001, 2004), Buhl (2003) have also analyzed this market). Given that liquidity is difficult to measure, we use different proxies to analyze the existence of commonality in liquidity. We first analyze whether there is evidence of factors that affect the liquidity of stocks jointly. We find evidence of the existence of three to four factors; these affect liquidity measures for all possible sizes of trades, not only for relatively small quantities that have been analyzed before. The explanatory power of the factors found is higher than what has been found in comparable studies on quote driven markets. This could be evidence that commonality not only exists under the different market structure of limit order markets, but also that it is of higher importance. Further, we find that the proportion of the variation of liquidity explained by the common factors varies throughout the day. This new finding adds to the intraday seasonalities found in previous research. It is also a hint that results might be misleading when liquidity proxies are used that are sampled at a certain time of the day. The statistical procedure we use does not shed light on the nature of the common factors. Therefore, we analyze in a next step the effect that several financial variables have on the liquidity of individual stocks. Average market liquidity explains a larger fraction of the variation of liquidity for the limit order market than for quote driven markets studied before (see e.g. Chordia et al. (2000a)), thus confirming the finding in the principal components analysis of a higher degree of commonality. Further, we analyze several additional financial variables on their effect on stocks liquidity. We find that market volatility has a strong impact. The remainder of this paper is organized as follows. Section 2 gives a review of the related literature. Section 3 describes the market structure of the SWX and the data used. Section 4 discusses the proxies for liquidity used. In section 5, the commonality in liquidity is analyzed. In section 6, we analyze financial variables having a market wide impact on stocks liquidity. section 7 concludes. 3

4 2 Related literature Liquidity is...a slippery and elusive concept, in part because it encompasses a number of transactional properties of markets (Kyle, 1985, p. 1316). Black (1971) mentions the following properties of a liquid market: There are always bid and asked prices for the investor who wants to buy or small amounts of stock immediately. The difference between the bid and asked prices (spread) is always small. An investor who is buying or selling a large amount of stock, in the absence of special information, can expect to do so over a long period of time at a price not very different, on average, from the current market price. An investor can buy or sell a large block of stock immediately, but at a premium or discount that depends on the size of the block. The larger the block, the larger the premium or discount. Based on these properties, Kyle (1985) defined the following aspects of a liquid market: Tightness: The cost of turning around a position over a short period of time. Depth: The size of an order flow innovation required to change prices a given amount. Resiliency: The speed with which prices recover from a random, uninformative shock. 2.1 Liquidity in quote driven markets Among the first contributions to the field of research on liquidity is the work on market microstructure (for on an overview on the market microstructure literature, see for example O Hara (1995), Biais et al. (2002)). O Hara (1995) defines market microstructure as the study of the process and outcomes of exchanging assets under explicit trading rules. Among the properties of liquidity investigated by this literature are the bid-ask spread and the depth that market makers post. Most of the empirical studies on liquidity have used data on quote driven markets such as the NYSE. On such a market, market makers quote prices and the quantities up to which they are willing to buy and sell at the quoted prices. Market makers thus provide a minimum level of liquidity to the market. Other market participants demand liquidity by placing market orders or provide 4

5 additional liquidity by placing limit orders. Order driven markets, on the other hand, function without market makers; liquidity is solely provided by investors placing limit orders. Several studies have shown that proxies for liquidity vary over time: In addition to deterministic variation such as intradaily or intraweekly variation of liquidity of individual stocks (Wood et al., 1985, Jain and Joh, 1988, Foster and Viswanathan, 1990, Engle and Lange, 1997), Chordia et al. (2001) also found unpredictable variation of market liquidity. Chordia et al. (2000b) have also found considerable cross-sectional heterogeneity in liquidity. As mentioned above, the empirical evidence on the existence of a stochastic factor that drives this variation over time of liquidity of all assets is ambiguous. Chordia et al. (2000a) look at quoted spreads, quoted depth and effective spreads of 1169 stocks traded on the NYSE throughout 254 trading days. Using transaction data, daily averages are calculated and market wide as well as industry wide averages are formed. Chordia et al. regress the first differences of the liquidity proxies against these market and industry averages and find that these aggregate quantities have a significant effect on the liquidity proxies in individual stocks, even after controlling for individual liquidity determinants, such as volatility, volume and price. They also find that the effect of market and industry averages is stronger on liquidity proxies calculated for portfolios. Huberman and Halka (2001) use as sample of 240 stocks traded at the NYSE over 254 days, selecting 60 stocks randomly from each size quartile. They consider the absolute and relative bid-ask spread as well as the quantity and dollar depth at noon. These liquidity proxies are averaged for each of the 4 portfolios of 60 stocks. Autoregressive processes are fitted to the resulting time-series. The authors interpret the presence of positive correlation in the residuals as evidence of a common factor affecting liquidity in different stocks. Hasbrouck and Seppi (2001) use liquidity proxies such as the spread, log spread, log size, quote slope and log quote slope, averaged over time intervals of 15 minutes length for 252 trading days for 30 Dow stocks. They perform a principal components analysis and find only little evidence of a common factor, the first common factor explaining 13 % of total variation of the log quote slope. If a common factor affects liquidity of all assets, it should be a priced factor. Various papers have investigated this question. Amihud and Mendelson (1986), Datar et al. (1998), Brennan and Subrahmanyam (1996), Brennan et al. (1998) find empirical evidence that cross-sectional differences in liquidity help to explain differences in expected returns. Amihud (2002) and Pástor and Stambaugh (2003) find that expected stock returns are cross-sectionally related to the sensitivities of returns to fluctuations in aggregate liquidity. Sadka (2003) finds that systematic liquidity risk, rather than the level of liquidity, is priced 5

6 for individual stocks. Acharya and Pedersen (2003) find that both the level and the risk of liquidity are priced. Gibson and Mougeout (2004) and Jones (2002) find that aggregate risk premia contain a premium for aggregate liquidity risk. 2.2 Liquidity in order driven markets A number of studies have investigated the properties of liquidity on limit order markets. Various studies have found that the order flow on such markets where traders face the choice between providing liquidity by placing a limit order or demanding liquidity by placing a market order react on changing market conditions (Biais et al., 1995, Harris and Hasbrouck, 1996, Al-Suhaibani and Kryzanowski, 2000, Griffiths et al., 2000, Ahn et al., 2001, Coppejans et al., 2001, Bae et al., 2003, Bloomfield et al., 2003, Ellul et al., 2003, Hollifield et al., 2003, Ranaldo, 2004). Irvine et al. (2000) have proposed a measure of illiquidity of a stock based on the order book, which they call the cost of a round trip, and investigated its ability to predict subsequent trading activity. The cost of a round trip is the cost of buying and selling immediately a position of a given size. As mentioned in the introduction, little work has been done on commonality in liquidity for order driven markets. Domowitz and Wang (2002) have used a measure of liquidity similar to the cost of a round trip proposed by Irvine et al. (2000). Their focus is on the relationship between order-type (market versus limit order), order-flow (buy versus sell orders) and their impact on commonality in liquidity and returns. They conjecture that liquidity commonality is related to co-movement of order-type, whereas return co-movement is due to co-movement of order-flow. In an empirical study, they look at the liquidity of 19 stocks from the ASX-20 index (Australian stock exchange) where six snap shots are taken on each trading day throughout They test the hypothesis whether order-type co-movement indeed leads to liquidity co-movement, and whether order-flow co-movement leads indeed to return co-movement by regressing for all pairs of stocks the correlations of returns (liquidity) on order flow and order type correlations. Their results support these hypotheses. Hansch (2003) has a different focus: He analyzes data on 100 stocks for August 2001 from the limit order book of Island, an electronic trading platform. On this platform, secondary trading takes place for stocks which are listed on AMEX, Nasdaq and NYSE; it is therefore an additional market place, but cannot be regarded as a pure limit order market, since the fact that the stocks are traded on other stocks exchanges where the stocks are listed will have a significant impact. Hansch uses a liquidity measure related to the cost of a round trip, based only on the fifteen best orders on each side of the book. 6

7 Similar to Hasbrouck and Seppi (2001), he performs a principal components analysis of this liquidity measure. He finds that the first principal component already explains 43% of the variance for the top volume stocks, which is orders of magnitude above what Hasbrouck and Seppi (2001) have found. In contrast, we use the complete order book of 19 stocks trade on the Swiss Stock Exchange (SWX) to analyze whether there are common factors affecting the liquidity of these stocks. Further, we also try to find financial variables that could explain this co-movement of liquidity. 7

8 3 Description of the market and data set 3.1 The market structure of the Swiss Stock Exchange SWX has the world s first fully automated trading platform (Swiss Stock Exchange, 2004e). It is a pure order driven market where all listed securities are permanently traded, with the option of voluntary market making (Swiss Stock Exchange, 2004d). The blue chips of the Swiss Market Index (SMI) which are the object of interest in this study are traded in the electronic virt-x system, based in London. The virt-x market, launched in June 2001, is based on an integrated trading, clearing and settlement model that allows to trade transnationally stocks from different European countries (Swiss Stock Exchange, 2004a). Electronic trading begins with the investor: participant banks investment advisors register incoming orders from their customers in their trading system. These data are forwarded to the trader and checked, or fed directly into the trading system by the trader. From here they go to the central exchange system of SWX, which acknowledges receipt of the order, assigns a time stamp to it and verifies its formal correctness. Depending on the type of transaction, the data are also transmitted to data vendors (Reuters, Telekurs, etc.). Trading orders are executed on a price/time priority, i.e. in the order of price (first priority) and time received (second priority) (Swiss Stock Exchange, 2004e). Trading hours are on business days from 09:00 to 17:30. Trading takes place continuously. Rules for trading halts are defined, similar to the circuit breakers at the NYSE, to avoid extreme market movements. Trading stops for 15 minutes (5 minutes for stocks priced less than CHF 10) if the potential follow up price deviates by 2% (25%) or more from the previous trade s or closing price (Swiss Stock Exchange, 2003b). During the last 10 minutes of the trading day, no orders are executed. Rather, they are matched in an auction at 17:30 when official closing prices are determined (Swiss Stock Exchange, 2004b). From 17:30 until 22:00 and again between 06:00 and 09:00, the so-called preopening takes place. Orders (bids and offers) may be entered or deleted in the electronic order book during pre-opening times, but no actual trades are made. A theoretical opening price is continuously calculated and displayed for the guidance of traders (Swiss Stock Exchange, 2004c). At the opening at 09:00, the opening price is determined and the orders are executed according to the matching rules of SWX. In order to establish the opening price at the start of trading (or upon resumption of trading after an interruption), the highestexecution principle is used; in other words, the price is fixed in such a manner as to achieve the largest possible turnover. If the potential opening price deviates by 2% (25% for stocks priced less than CHF 10) or more from the reference 8

9 price (which equals essentially the previous traded price), opening is delayed by 15 minutes (5 minutes) (Swiss Stock Exchange, 2003b). All equity orders with a size of less than CHF must be executed within official trading hours through the SWX trading system, whereas orders above this size limit may be traded off-exchange (Table 4 reports the importance of the off-exchange trades). However, all off-exchange transactions must be reported within 30 minutes of their conclusion (Swiss Stock Exchange, 2004d). The tick size varies with the stock price (Swiss Stock Exchange, 2003b) according to the rules summarized in Table 1. This implies that when large changes in the stock price take place, tick sizes will change and hence also the observed spreads. This is different from other markets such as the NYSE where the tick size remains fixed. Besides limit and market orders, there are three further types of orders (Swiss Stock Exchange, 2003a): The hidden size order is a large order whose size is in the order book is only partially visible; all participants see that this order is of hidden size. The accept order is executed immediately against all orders in the book, either fully or partially. The remaining order is cancelled immediately and the order is never entered in the order book. The fill or kill order is like an accept order which is only executed if full matching is possible. 3.2 Description of the data set The data set consists of all orders that have been submitted on the SWX between May and July 2002 together with all trades during this period. For each order, the time of submission is given, the state of the order book at the time of submission (trading, opening, stop trading, etc.), the original size and price of the order, the type and direction (buy or sell) of the order, the quantity and time at which part of the order was executed, the expiry period, the time and reason of actual deletion and the state of the order book at the time of deletion. Only stocks that are part of the Swiss Market Index (SMI) are considered. 1 Table 2 gives an overview of the stocks in the sample. The data available 1 The SMI is a capital-weighted, non-dividend-corrected index (see products/indices/products_smi_en.html). It comprises at maximum 30 of the most significant equity-securities issues included in the Swiss performance index (SPI) which is a dividend-corrected index including all equity traded at the SWX of companies domiciled in Switzerland or Liechtenstein. The SMI is adjusted on the basis of the free float market capitalization. An issuer s market capitalization is adjusted to reflect the number of shares in fixed ownership, whereas only the freely tradable portion of the outstanding shares is taken into account. 9

10 cover 66 trading days from May 3rd to July 31st The sample only covers about 45% of the market capitalization of the SMI. 2 It includes with NOVN the stock with the highest market capitalization traded on the SWX. As for the market environment during the observation period, the Swiss market experienced during the second quarter 2002 a significant downturn. Unfortunately, this is likely to have an impact on our results and it would be very interesting to have additional data on a different time period. Figure 1 displays the SMI during this period. The total market value of the stocks in the sample declined from 349 BCHF to 289 BCHF (see Table 2). This downturn affected all stocks in the sample: The closing price of each stock on July 31st is below its level on May 3rd. Also the mean daily return, given in Table 3, is negative for all stocks and zero for a single stock. Another indication of the turbulences during the sample period are the large standard deviations of the returns, ranging from 1.2 to 4.4. The data set comprises trades (see Table 4). 99% of the trades have been carried out on the exchange. The percentage of trades on exchange does not vary across stocks. The trades on the exchange make up only 76% of the 1.3 billion shares traded. The importance of off exchange trading varies considerably across stocks, it ranges between 5.4% and 66.0% of the total traded volume. The variation across stocks of the median turnover of off exchange trades (shown in columns Reported trades and Confirmed trades of Table 4) is relatively small, while the variation of the mean turnover is considerable. This indicates that stocks with a very low on exchange trading volume experienced during the observation period a small number of off exchange trades of large volume. Further, the data set comprises orders (see Table 5). 90.3% of all orders are limit orders, 8.2% are market orders and 1.5% are special orders, that is hidden, accept or fill or kill orders. Stocks are quite homogeneous with respect to the distribution of order types. 50.7% of all orders are sell orders. This imbalance reflects the downward pressure on the stock prices. Limit orders are evenly split into buy and sell orders. Yet, several stocks show a very high fraction of market orders being sell orders. The contrary is true for the special orders where a higher fraction are buying orders. As shown in Table 5, the chances that a limit order would be executed at least partially amounts to 48.6% on average, ranging between 33.6% and 56.6% for the different stocks. From these limit orders with at least partial execution, 19.0% have been fully executed at the instant of submission, like market orders, and have never been entered in the book as open orders. Another part of the limit orders, 25.5%, have held for some time the status as open orders and have 2 The data on the remaining stocks in the SMI still needs to be produced by the SWX. 10

11 eventually been fully executed. Only 3.8% of all limit orders were deleted upon request of the trader and 0.2% of all orders were only partially executed before their expiry date. The majority of the orders were not traded against. 46.9% were deleted upon request of the trader, where this fraction varied between 37.3% and 63.2% among stocks. A further 4.5% of all limit orders expired without any trade. The last column of Table 5 gives the fraction of all orders that were only valid for a single day, that is, the number of orders which expired on the same day as they were submitted on. The vast majority, 84.0%, of all orders were submitted for a single trading session only. Table 6 gives an overview of the special orders. They are mostly (86.8%) accept orders and 11.3% are of the type fill or kill. Almost all (75.6%) accept orders got fully or partially matched. This suggests that accept orders are very similarly used as market orders. The same is true for fill or kill orders: 10.2% out of the total of 11.3% of all special orders that were of this type got matched. Table 7 presents some descriptive statistics on the limit and market orders submitted on average on a trading day. Both the number of limit and market orders submitted on a day vary considerably, as the corresponding minima and maxima show. Also the daily turnover of the submitted limit orders displays large variation, the minimal turnover ranging between 0.4% and 3% and the maximal turnover ranging between 2.3% and 14%. As for the turnover of the submitted market orders, the minimal turnover is between 0.004% and 0.05% and the maximal turnover is between 0.38% and 1.1%. 3.3 Selection of the sample and construction of the order book The database contains information on the time of the order submission, its original size and price, all the partial executions of the order and the time of deletion. It contains all orders submitted between May 3rd and July 31st In order to construct the order book for any time period, we need to know the orders which have been submitted before this period and that have not been executed before the start of this relevant period. We therefore start only to build the order book as per May 10th, thus allowing orders to be submitted between May 3rd and May 9th without execution. This initial period of five trading days is sufficiently long, given that a large fraction of the orders expires at the end of the trading day of their submission as can be seen from the last column in Table 5: 84.0 % of all orders were only valid a single day. In addition, Table 8 gives an overview of the time that orders remained in the book: After 11

12 24 hours, more than 98% of all orders have been removed from the book. From this, we consider for the analysis a time period starting May 10th and ending July 31st 2002, spanning 59 trading days. To reconstruct the order book, only orders that have been visible in the book are used, that is, hidden, accept and fill or kill orders are omitted. These special orders only make up 1.5% of all limit orders (see Table 5), thus the loss of sample size is not severe. Further, as shown in Table 6, 87% of the special orders were accept orders that function like market orders: They are executed against all orders in the book and then immediately deleted. They are therefore not part of the liquidity supply. The order book is known at each instant. As customary in much of the literature on high-frequency data, we use snapshots taken at time intervals of 5 minutes, between 09:05 and 17:15. This gives 99 observations a day, T = 5841 in total. Observation times are denoted by t i, i = 1, 2,..., T, at these intervals of 5 minutes, between 09:05 and 17:15, starting May 10th and ending July 31st At various times, trading in one or several of the stocks in the sample has been stopped according to the rules of the SWX described in section 3.1. The trading halts lasted typically for 15 minutes, some up to an hour. Table 13 shows an overview how on number of trading halts of the different stocks. For such trading halts, the liquidity proxies defined in the section were set equal to the their value at the nearest time at which trading took place. It appears to be likely that this interpolation will not have a significant effect on the analysis, given that for most stocks, less then 50 trading halts were observed out of a total number of T = 5841 observations. For each point of time t i, the total amount of shares on offer for buying or selling defines supply curves of liquidity, such as the example shown in Figure 2, as follows: At any time t i, N A t i orders are open on the ask side and N B t i on the bid side. The n th order on side s S {A, B} open at time t is characterized by its price p s t,n and quantity q s t,n. The orders are sorted from best to worst, that is, such that p A t,n p A t,n+1, n = 1, 2,..., Nt A 1 on the ask side and p B t,n p B t,n+1, n = 1, 2,..., Nt B 1. The total depth of any side of the book is then the sum of all quantities offered, N s t Q s t qt,n. s (1) n=1 The quantity offered by the n best orders is defined as Q s t,n n qt,m, s s S. (2) m=1 12

13 Based on the orders, two forms of the liquidity supply curve can be derived. The first is the supply curve such as shown in Figure 5(b), Q s t (p) Q s t,n s (p), 0 p ps t,n, s S, where t s N A (p) sup{n N A t N B (p) sup{n N B t : p A t,n p}, : p B t,n p}. For any price p, Q s t (p) equals the total number of shares offered at this price. Alternatively, the supply curve can also be defined in its inverse form, such as shown in Figure 5(a): Pt s (Q) p s t,n s (Q), 0 Q Q s t where N N s (Q) inf{n N t : qt,n s Q}. P s t (Q) gives the price that a market order of size Q has to pay/receives for the last unit. n=1 The price per share for an order of size Q is then given by p s t (Q) N s (Q) 1 n=1 p s t,nqt,n s + (Q Q t,n s (Q) 1)p t,n s (Q) /Q, 0 Q Q s t. When an order of size Q is to be matched, the offer walks up the book for the first N s (Q) 1 orders; the remaining quantity Q Q t,n s (Q) 1 is then executed at the price p t,n s (Q). From the price schedules on the ask and bid side, the mid-price is defined as the average of the best bid price p B t and best ask price p A t, p mid t 1 ( p A 2 t + p B ) t. In order to attain comparable quantities, quantities are always expressed as a fraction of the free float of the respective stock. Since it is only the market capitalization of the free floating stocks that is relevant for the stocks weighting in the SMI, SWX has to collect the information on the free float. Free float describes that portion of a given joint-stock company s shares that is not closely held as measured against the total number of issued shares (Swiss Stock Exchange, 2004). Holdings by a group of persons of five percent or more of the outstanding shares is considered as being closely held. The numbers on freefloat published by the SWX on the stocks in the sample are given in Table 2. We standardize quantities by the number of shares free floating, given by the prod- 13

14 uct of the total number of outstanding shares NO and the fraction of free float F F l(t) at time t, Q F F l(t)no. Both of these quantities are given in Table 2. This way of standardizing is motivated by the large differences in the free float as shown in Table 2: Some stocks, such as the former state owned Swisscom (SCMN), have a ratio of free float as little as one third of all shares while other stocks have a free float of 100%. The figures on the number of outstanding securities reported in Table 2 further show that standardization of quantities is essential since these numbers display large variation across stocks. Price levels of the stocks in the sample vary considerably, as shown in Table 2. Therefore, wherever the structure of the order book is of interest and not the intertemporal behavior of prices, prices are expressed as multiples of the mid-price. Figures 2(b) and 2(d) show an example of such a standardized order book, both the supply curve and the inverse supply curve. 14

15 4 Proxies of liquidity 4.1 Spread of best bid-ask prices In quote driven markets, tightness of the market can only be observed at the quoted prices and quantities as spread between the best bid and ask price; depth is only observable as the quantities up to which the quoted prices of the market maker are valid. Since a large part of the literature has focused on these measures, we will briefly discuss them. Table 14 shows descriptive statistics on the observed spreads. Relative spreads lend themselves better for comparison across stocks that are traded at very different price levels and therefore have different tick sizes and thus minimal spreads. The average spread across all stocks was 0.28%. The fact that the median of 0.23% is smaller than the mean indicates that the distribution of the spreads is skewed. This is confirmed by the 99% quantile which equals 0.89%. The variation of the median spread across stocks is quite considerable, ranging from 0.08% to 0.39%. At the 99% quantile, this spreads range between 0.27% and 1.51%. Table 14 shows that the market is much more liquid for the largest stock in the sample, NOVN: The quantiles of the absolute spread imply that the spread is most of the time at its minimum, as determined by the tick size for this stock which amounts to CHF The rules of the SWX concerning the tick size which have been described in section 3.1 pose a significant difficulty for the statistical analysis of the spread data. The range of possible values that the spread can attain may change due to price movements. Figure 3 shows an example: Before July 15th 2002, BALN was traded at a price above 100 CHF and the tick size therefore was 0.25 CHF (see Table 1). When the stock price dropped on July 15th 2002 below 100 CHF, tick size was reduced to 0.05 CHF. In total, this happened to three stocks in the sample. This is also documented by the last two columns in Table 14 that report the number of different values that the spread has attained during the full period from May to July (05-07) and during the subperiod from May to June (05-06). Therefore, any statistical analysis of the spread needs not only to handle the discreteness of the spread data, but also the apparent structural break that happened in the last third of the sample period. Further, when the stock price crosses one of the barriers defining the tick size in table Table 1, the spread will increase or decrease without any change in market liquidity. This is in sharp contrast to other stock exchanges such as the NYSE where the tick size, and thus the range of possible spreads, is fixed and the constant whatever the price level is. Yet, under the rules of the SWX the spread of the best bid and 15

16 ask price is not a convenient quantity due to economic and statistical reasons and we do not further include it in our analysis. A further quantity often used in the analysis of the liquidity of quote driven markets is the effective spread, defined as twice the absolute value of the difference between the trading price and the mid-price prevailing at the time of price. In quote driven markets, this variable contains additional information since market makers may execute trades at prices between their quoted bid and ask prices. In the case of a pure limit order market, such trades cannot take place: At each trade, a limit order is matched on either side of the book, either against a new limit order or a market order. Therefore the trading price always corresponds to the best price on the opposite side of the order book. 4.2 Depth related liquidity measures A defining feature of market liquidity is the degree to which large quantities can be traded. Therefore, the depth of the book measures the quantity dimension of liquidity: A large depth is necessary but not sufficient for market to be liquid. We discuss three different depth measures in this section that will be used in the empirical analysis. In order to have comparable units, depth is always measured in millionths of the total free float Depth of best orders Depth of the best bid and ask orders, denoted by Q s t,1 s S as defined in (2), are widely used measures of depth in quote driven markets. The market maker guarantees to buy or sell any quantity up to the quoted depth. In the sense of Kyle (1985), this is the trade size required to change the price, because up to the quoted depth, market makers trade at the quoted prices. In a pure limit order driven market, the economic significance of the depth at the best orders is less clear: It can happen, that an order has been partially executed up to a single share and that this order happens to be the best order on its side of the book. The depth of the best order will then be only one share, while large quantities might be offered by the next best orders at the next tick. The depth of the best orders will thus be an incomplete measure of the depth right at the mid-price. A further problematic consequence of this fact is that the depth is highly volatile. Figure 4 shows in the upper (lower) panel the maximum (minimum) of the number of shares available at the best bid and ask offer on a single day, averaged across all stocks. It can be seen that the depth of the best bid and ask offers, even for the sample average, undergoes dramatic changes within a day, moving from its minimum to its maximum. On May 27th 2002, for example, 16

17 the minimum depth was 37 shares, while the maximum was attained at 6441 shares. Given that the quantity of the currently best offers on both sides of the book vary tremendously, previous research has used different measures of depth, such as the depth at the best 5 quotes (Biais et al., 1995, Al-Suhaibani and Kryzanowski, 2000, Ahn et al., 2001) or depth at a number of ticks away from the current mid-price (Coppejans et al., 2001). We follow the literature and calculate the depth Q s t,5 that the best five orders on each side of the book offer (see the definition (2)). Table 17 reports the descriptive statistics for the depth Q s t,5. The minimal depth is on average % of the shares in the freefloat, while the mean depth is 0.008% averaged over all stocks. Interestingly, both the median and mean depth are very similar on both sides of the book. Thus, under normal market conditions (as mentioned, one needs to keep in mind that the market experienced during the sample period a significant downturn), the depth provided on the buying and selling side behave very similarly Total depth of the book Another measure of liquidity is the total amount of stocks that are in the limit book available, i.e. the total depth Q s t, s S of the book, as defined in (1). Table 15 reports some descriptive statistics on the total depth of either side of the book. On average, the open orders correspond to about 0.2% of the free float on the ask side and to about 0.1% on the bid side. This surplus of selling orders reflects the general downward tendency of the market during the given period. In contrast, the best five orders, considered above, provide on average only a depth Q s t,5 of 0.008%. Yet, the descriptive statistics on Q s t,5 are symmetric on both sides of the book. Thus, the order imbalance reflected by the differing total depth Q s t, s S on each side of the book is caused by a large number of selling orders that have limit prices far away from the current price level. The total depth of the book shows huge variation on both sides of the book: For several stocks, the maximum depth amounts to about ten to twenty times the minimal depth. This is a first indication that the quantity dimension of liquidity displays large changes over time. To get an idea about the economic relevance of this variation in quantity offered by the book, we ask whether depth was at all times sufficient to execute the largest market orders. Therefore, we compare the quantiles of size distribution of the market orders reported in Table 10 with the minimal total depth of the book ever observed. 99% of the market orders, as reported in the last column of Table 10, could be settled at all times, except for three stocks 17

18 (BAER, BALN, CFR). Thus, total depth of the book varies around levels much higher than most market orders ever placed Depth at a premium around the mid-price Kyle s 1985 original definition of depth refers to the quantity that needs to be traded to move the price by a given amount. We will consider the quantity required to change the price by 1%. To this end, we consider the depth Q A t (1 + 1%) of the book available at the mid-price plus 1% on the ask and the analogous quantity Q B t (1 1%) available at a 1% discount on the bid side, respectively. Table 16 reports the descriptive statistics for the depth available at a premium of 1%. It should be noted that several stocks have experienced times where the spread was larger than 2% (see in Table 14 the columns on the maximum and the 99% quantile of the relative spread) and therefore no shares were offered at the mid-price plus or minus 1%. This explains why the minimum depth Q s t (1 ± 1%) is zero for most stocks. Table 16 reports several quantiles from the distribution of the depth available at a premium of 1%. As before, we compare this depth with the size of market orders to get an idea of its economic significance. We find that very large market orders, as given by the 90% quantile of the market orders size in Table 10, could be executed on both sides of the book at least 95% of the time, moving the price by less than 1% (that is, the 90% quantile of the market order size, given in Table 10, is less than the 5% quantile of Q s t (1 ± 1%), given in Table 16). Yet, at least 1% of the time, the 10% largest market orders had an impact on price of more than 1% (that is, the 90% quantile of the size of the market orders, given in Table 10, is larger than the 1% quantile of the Q s t (1 ± 1%), given in Table 16). And 5% of all times, the 1% largest market orders moved prices by more than 1% (that is, the 99% quantile of the size of the market orders, given in Table 10, is larger than the 5% quantile of the Q s t (1 ± 1%), given in Table 16). We conclude that relatively often, large orders have a significant price impact. It would be interesting to study whether the depth was especially low at times of market turmoil, and thus pose an additional risk at such a time. As for the depth Q s t,5 of the best five offers, both the numbers for median and mean depth are very similar, indicating similar depth on the buy and sell side under normal market conditions. 4.3 Cost of illiquidity Black (1971) has pointed out that in a liquid market, traders need to accept a premium or discount in order to trade a large order immediately. The size of this premium (for brevity, we call also the discount required on the bid side 18

19 premium) will increase with order size and will be higher at times where the market is less liquid. Therefore, we use the premium for a given order size as a proxy for market liquidity. We call the difference between the price per share offered by the limit orders and the mid-price the cost of illiquidity (CIL) of the respective side of the book (see Irvine et al. (2000)), l s t (Q) p s t (Q) p mid t, s S. It is the cost that a trader faces who requires immediate trading by placing a market order of size Q. If the market was perfectly liquid, that is, the spread was zero and depth equal to all shares that are in the free float, the CIL amounted to zero for any order size. On the other hand, if the market is not perfectly liquid, the CIL for a quantity Q exceeding the total depth of the book cannot be observed and would be needed to set equal to infinity. To avoid such difficulties, the analysis in this paper will always be restricted to quantities smaller than the total depth of the book. This measure has been proposed by Irvine et al. (2000) (this measure was also used by Domowitz and Wang (2002) and Hansch (2003)) who analyzed the cost of a round trip, that is, buying and selling a quantity at the same instant. This corresponds to the sum of the CIL of the ask and bid side, l A t (Q) + l B t (Q). CIL is a natural generalization of the best bid-ask spread to the situation where the complete order book is known: Where the latter amounts to twice the premium that has to be accepted, relative to the mid price, in order to trade a single unit, the CIL indicates the premium relative to the mid-price that has to be accepted when trading any desired quantity. l s t (Q) depends on the trade size Q. Different trade sizes have been used in the literature: Irvine et al. (2000) used 5 different trade sizes, ranging from 5000 CAD up to CAD; Hansch (2003) uses multiplies of the average trade size; and Domowitz and Wang (2002) used trading quantities ranging from 1000 to shares. We consider the following four hypothetic order sizes for which the CIL will be calculated: Small orders of size Q Sm, corresponding to the first quartile of the size of market orders, given in Table 10. Median orders of size Q Md, corresponding to the median of the size of market orders. Order size Q MinD, equal to the minimum depth that has ever been observed in a given stock on either side of the book, Q MinD min( min i=1,...t { Q A t i }, min i=1,...t { Q B t i }). 19

20 Order size Q Gmin, equal to the minimum depth that has ever been observed for any stock (that is, the minimum of Q MinD of all stocks) shown in Table 15. Small and median orders reflect the size of actually placed market orders. The CIL for these orders sizes indicate the cost that average and below average sized orders face. Yet, observed market orders are only one part of the demand for liquidity: Since market participants will only place such orders that appear favorable to them in view of the liquidity of the market, we do not observe the largest orders because the market conditions are so unfavorable that they are not placed. Yet, we would like to measure also the liquidity for these unobservable demand for liquidity. We therefore calculate in addition the CIL for two very large hypothetic order sizes: Q MinD amounts for any stock to the minimal total depth ever observed in the book for this title. This is the largest quantity for which the data allows us to calculate the CIL for each point of time. As shown in Table 18, Q MinD varies considerably across stocks. To measure the CIL for a hypothetic large order of constant size across different stocks, we use the fourth quantity, Q Gmin. This quantity equals the minimum of Q MinD, which is in our sample equal to the minimal depth of the stock BALN. Table 19 report the minimum, maximum and mean CIL both for the ask and bid side for the different stocks. Since the premium that a trader has to accept increases with order size, the minimum CIL is constant for all order sizes. On average, the minimum CIL amounted to 4.7 basis points. This corresponds quite closely to one half of the average minimum spread reported in Table 14, suggesting that during the most liquid times, the only costs of immediacy faced by traders was the spread. Yet, both the maximum and the mean CIL reveal that the spread by far underestimates the costs of immediacy: The means for small and median orders are about three times as high as the half-spread. Interestingly, the mean values of the CIL are very similar on both sides of the book, suggesting similar costs of immediacy on average on both sides. Both the maximum values for the CIL and the quantiles (not shown for the sake of space) suggest that the CIL is very high at certain times. Again, it would be interesting to study whether the costs are high at times of market turmoil, and thus increase the risk of positions in the stock even further. In general, the CIL for small and medium sized orders is below 1%, on average at a level of about 0.14%. The CIL thus constitute a second order risk compared to the market risk where standard deviations of daily returns are in the range from 1% to 4%, as reported in Table 3. Yet, for trading strategies with a high trading frequency, the CIL can be of concern. 20

21 The summary statistics on the CIL for the largest stock in the sample, NOVN, confirm the higher liquidity in this stock: For small trading quantities (Q Sm and Q Md ), the mean CIL amount to 5 basis points only on both sides of the book, while these numbers are for most other stocks above 100 basis points. 5 Commonality in liquidity Research on commonality has proceeded mainly along three paths. Three main approaches have been used in the. Chordia et al. have regressed the liquidity proxies against market and industry averages. these regressions are interpreted as evidence for commonality. The significant coefficients in Huberman and Halka (2001) have looked at the residuals of autoregressive processes that were fitted to the time-series of the liquidity proxies. The presence of positive correlation in these residuals is interpreted as evidence of a common factor affecting liquidity in different stocks. And Hasbrouck and Seppi (2001) have performed a principal components analysis of the liquidity proxies. We follow this last approach. Principal components analysis looks for linear combinations of the data that explains as much as possible of the variance of the data. The weights of the variables in the linear combination forming the first principal components correspond to the first eigenvector of the variables covariance matrix; the weights of for the second principal components correspond to the second eigenvector of the covariance matrix and so on (for a rigorous treatment of principal components analysis, see e.g. Flury (1988), Krzanowski (1988), Rao (1996)). Since principal components analysis is sensitive to units of measurement, the data is usually standardized to unit variances. Equivalently, the correlation matrix is used. Total variance in this case is just equal to the number of variables in the analysis. The ratio of the total variance explained by the i th principal component is then given by v i λ i n, (3) where λ i is the i th eigenvector of the correlation matrix of n variables. We denote by m i=1 V m λ i n the ratio of the total variance explained by the first m principal components jointly. Under the assumption of normality for the original variables, then the sample eigenvalues ˆλ i are asymptotically independent and normally distributed (4) 21